On homogeneous chaos

1991 ◽  
Vol 110 (2) ◽  
pp. 353-363 ◽  
Author(s):  
Nigel Cutland ◽  
Siu-Ah Ng
Keyword(s):  
Brownian Motion ◽  
Wiener Space ◽  
Chaos Decomposition ◽  
Image Position

AbstractThis paper discusses the Wiener–Itô chaos decomposition of an L2 function φ over Wiener space, and is concerned in particular with the identification of the integrands ƒn in the chaos decompositionFirst these are identified as Radon–Nikodým derivatives. Two elementary non-standard proofs of the Wiener–Itô chaos decomposition are given, based on Anderson's construction of Brownian motion and Itô integration.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (04) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτ E (max0≤t≤τ X t − c τ), where X = (X t ) t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | X t = g * (max0≤t≤s X s )} where s ↦ g *(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g *(s) ∼ ((Δ − 1) / K Δ)1 / Δ s 1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (sup t>0 X t ) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

A limit theorem for certain disordered random systems

1994 ◽  
Vol 26 (04) ◽  
pp. 1022-1043 ◽  
Author(s):  
Xinhong Ding
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Limit Theorem ◽  
Random Systems ◽  
Joint Probability ◽  
Dynamical Behaviour ◽  
One Dimensional ◽  
Linear Stochastic Differential Equation ◽  
The One ◽  
Image Position

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in : where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝ p -valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation where P(t, dx dy) is the joint probability law of ξ and X(t).

Biharmonic functions and Brownian motion

1967 ◽  
Vol 4 (1) ◽  
pp. 130-136 ◽  
Author(s):  
L. L. Helms
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Boundary Function ◽  
Normal Derivative ◽  
Dimensional Euclidean Space ◽  
Biharmonic Functions ◽  
Neumann Type ◽  
Bounded Open Subset ◽  
Image Position ◽  
First Time

Let R be a bounded open subset of N-dimensional Euclidean space EN,N ≧ 1, let {xt: t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex[g(xτ)] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δkg(XΔ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1u = Δ(Δku) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].

On the zeros and Fourier transforms of entire functions in the Paley–Wiener space

1996 ◽  
Vol 119 (2) ◽  
pp. 357-362 ◽  
Author(s):  
Konstantin M. Dyakonov
Keyword(s):  
Entire Function ◽  
Complex Number ◽  
Compact Support ◽  
Fourier Transforms ◽  
Entire Functions ◽  
Wiener Space ◽  
Dirichlet Integral ◽  
Fixed Complex ◽  
Image Position

AbstractLet f be an entire function of the formwhere ø is a function in L2(ℝ) with compact support. If f|ℝ is real-valued then, for obvious reasons, (a) the supporting interval for ø is symmetric with respect to the origin, andAssuming that f has no zeros in {Im z > 0}, we prove that the converse is also true: (a) and (b) together imply that f|ℝ takes values in αℝ, where α is a fixed complex number.The proof relies on a certain formula involving the Dirichlet integral, which may be interesting on its own.

Optimal stopping and maximal inequalities for geometric Brownian motion

1998 ◽  
Vol 35 (4) ◽  
pp. 856-872 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peskir
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Stopping Time ◽  
Practical Interest ◽  
Maximal Inequality ◽  
Geometric Brownian Motion ◽  
Stopping Times ◽  
Image Position

Explicit formulas are found for the payoff and the optimal stopping strategy of the optimal stopping problem supτE (max0≤t≤τXt − c τ), where X = (Xt)t≥0 is geometric Brownian motion with drift μ and volatility σ > 0, and the supremum is taken over all stopping times for X. The payoff is shown to be finite, if and only if μ < 0. The optimal stopping time is given by τ* = inf {t > 0 | Xt = g* (max0≤t≤sXs)} where s ↦ g*(s) is the maximal solution of the (nonlinear) differential equation under the condition 0 < g(s) < s, where Δ = 1 − 2μ / σ2 and K = Δ σ2 / 2c. The estimate is established g*(s) ∼ ((Δ − 1) / K Δ)1 / Δs1−1/Δ as s → ∞. Applying these results we prove the following maximal inequality: where τ may be any stopping time for X. This extends the well-known identity E (supt>0Xt) = 1 − (σ 2 / 2 μ) and is shown to be sharp. The method of proof relies upon a smooth pasting guess (for the Stephan problem with moving boundary) and the Itô–Tanaka formula (being applied two-dimensionally). The key point and main novelty in our approach is the maximality principle for the moving boundary (the optimal stopping boundary is the maximal solution of the differential equation obtained by a smooth pasting guess). We think that this principle is by itself of theoretical and practical interest.

scholarly journals Polymer measure: Varadhan's renormalization revisited

2015 ◽  
Vol 27 (03) ◽  
pp. 1550009 ◽  
Author(s):  
Wolfgang Bock ◽  
Maria João Oliveira ◽  
José Luís da Silva ◽  
Ludwig Streit
Keyword(s):  
Brownian Motion ◽  
Rate Of Convergence ◽  
Local Time ◽  
Intersection Local Time ◽  
Planar Brownian Motion ◽  
Chaos Decomposition

Through chaos decomposition, we improve the Varadhan estimate for the rate of convergence of the centered approximate self-intersection local time of planar Brownian motion.

scholarly journals Approximation by Brownian motion for Gibbs measures and flows under a function

1984 ◽  
Vol 4 (4) ◽  
pp. 541-552 ◽  
Author(s):  
Manfred Denker ◽  
Walter Philipp
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Gibbs Measures ◽  
Hölder Continuous ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Image Position ◽  
Class Of Functions

AbstractLet denote a flow built under a Hölder-continuous function l over the base (Σ, μ) where Σ is a topological Markov chain and μ some (ψ-mining) Gibbs measure. For a certain class of functions f with finite 2 + δ-moments it is shown that there exists a Brownian motion B(t) with respect to μ and σ2 > 0 such that μ-a.e.for some 0 < λ < 5δ/588. One can also approximate in the same way by a Brownian motion B*(t) with respect to the probability . From this, the central limit theorem, the weak invariance principle, the law of the iterated logarithm and related probabilistic results follow immediately. In particular, the result of Ratner ([6]) is extended.

A limit theorem for certain disordered random systems

1994 ◽  
Vol 26 (4) ◽  
pp. 1022-1043 ◽  
Author(s):  
Xinhong Ding
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Limit Theorem ◽  
Random Systems ◽  
Joint Probability ◽  
Dynamical Behaviour ◽  
One Dimensional ◽  
Linear Stochastic Differential Equation ◽  
The One ◽  
Image Position

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in : where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝp-valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation where P(t, dx dy) is the joint probability law of ξ and X(t).

On the Hausdorff dimension of Brownian cone points

1985 ◽  
Vol 98 (2) ◽  
pp. 343-353 ◽  
Author(s):  
Steven N. Evans
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Two Dimensional ◽  
Image Position

Let B(t) be a two-dimensional Brownian motion. For 0 < α < 2π, setand, for 0 ≥ β< 2π, let F(α,β) be F(α) rotated through an angle β about the origin.

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